U.S. patent application number 11/570896 was filed with the patent office on 2008-11-06 for imaging method with back projection.
This patent application is currently assigned to KONINKLIJKE PHILIPS ELECTRONICS N.V.. Invention is credited to Thomas Koehler, Tim Nielsen, Roland Proksa, Andy Ziegler.
Application Number | 20080273655 11/570896 |
Document ID | / |
Family ID | 35782170 |
Filed Date | 2008-11-06 |
United States Patent
Application |
20080273655 |
Kind Code |
A1 |
Nielsen; Tim ; et
al. |
November 6, 2008 |
Imaging Method with Back Projection
Abstract
The invention relates to an imaging method, especially a
computerized tomography method, with which an object is penetrated
by rays from different directions and measured values, which depend
upon the intensity of the rays after penetrating the object, are
acquired by a detector unit. From these measured values, an object
image is reconstructed by means of back projection of
measured-value-dependent back projection values. Therein, the
object image is divided into overlapping, quasi-spherically
symmetric image segments, each being defined by an image value and
a quasi-spherically symmetric base function. Furthermore, during
the back projection, the back projection values are added in
proportions to the image values, wherein the proportion of a back
projection value, which is added during the back projection to an
image value, is dependent on a proportionality factor, which is
equal to the average value of the line integrals of the base
function belonging to the respective image value along those rays
that have generated the measured value, on which the respective
back projection value is dependent.
Inventors: |
Nielsen; Tim; (Hamburg,
DE) ; Ziegler; Andy; (Hamburg, DE) ; Koehler;
Thomas; (Norderstedt, DE) ; Proksa; Roland;
(Hamburg, DE) |
Correspondence
Address: |
PHILIPS INTELLECTUAL PROPERTY & STANDARDS
595 MINER ROAD
CLEVELAND
OH
44143
US
|
Assignee: |
KONINKLIJKE PHILIPS ELECTRONICS
N.V.
Eindhoven
NL
|
Family ID: |
35782170 |
Appl. No.: |
11/570896 |
Filed: |
June 24, 2005 |
PCT Filed: |
June 24, 2005 |
PCT NO: |
PCT/IB2005/052090 |
371 Date: |
December 19, 2006 |
Current U.S.
Class: |
378/19 ;
382/131 |
Current CPC
Class: |
G06T 2211/421 20130101;
Y10S 378/901 20130101; A61B 6/027 20130101; G06T 11/006
20130101 |
Class at
Publication: |
378/19 ;
382/131 |
International
Class: |
A61B 6/00 20060101
A61B006/00; G06K 9/00 20060101 G06K009/00 |
Foreign Application Data
Date |
Code |
Application Number |
Jun 25, 2004 |
EP |
04102956.2 |
Claims
1. An imaging method, in particular a computerized tomography
method, comprising: reconstruction of an object image by back
projection of back projection values p.sub.j-{tilde over (p)}.sub.j
that depend on measured values, wherein the object image is divided
into overlapping, quasi-spherically symmetric image segments, each
one of which is defined by an image value (c.sub.i) and a
quasi-spherically symmetric base function b.sub.i(x), wherein the
back projection values (p.sub.j-{tilde over (p)}.sub.j) are added
in proportions to the image values (c.sub.i) during the back
projection.
2. An imaging method as claimed in claim 1, characterized in that
the proportion of a back projection value (p.sub.j-{tilde over
(p)}.sub.j), which is added to an image value (c.sub.i) during the
back projection, is dependent on a proportionality factor
(w.sub.ji), which is equal to the average value of the line
integrals of the base function (b.sub.i(x)) belonging to the
respective image value (c.sub.i) along those rays that have
generated the measured value (p.sub.j) on which the respective back
projection value (p.sub.j-{tilde over (p)}.sub.j) is dependent.
3. An imaging method as claimed in claim 1, characterized in that
the proportion of a back projection value (p.sub.j-{tilde over
(p)}.sub.j), which is added during the back projection to an image
value (c.sub.i) is proportional to the respective proportionality
factor (w.sub.ji).
4. An imaging method as claimed in claim 1, characterized in that
each detector element (51) has a rectangular area and that the
proportionality factor (w.sub.ji) for a back projection value
(p.sub.j-{tilde over (p)}.sub.j) and an image value (c.sub.i) is
determined with the following steps: providing a table, in which
each combination of radiation source position, detector element and
image segment is assigned one or more sub-areas (T1 . . . T17) and
average area values assigned to these sub-areas (T1 . . . T17),
wherein each sub-area (T1 . . . T17) is rectangular and arranged in
a such a way that a corner of the respective sub-area (T1 . . .
T17) coincides with a center (53) of the base function (b.sub.i(x))
of the image segment, which center (53) is projected on the
detector area (18) along rays emanating from the radiation source
position that is assigned to the respective sub-area (T1 . . .
T17), and that a diagonally opposite corner of the respective
sub-area (T1 . . . T17) coincides with a corner of the rectangular
area of the detector element (51) to which the respective sub-area
(T1 . . . T17) is assigned, and wherein the average area value
assigned to a sub-area (T1 . . . t17) is equal to the average value
of the line integrals of the base function (b.sub.i(x)) of the
image segment, to which the respective sub-area (T1 . . . T17) is
assigned, along those rays that strike the respective sub-area (T1
. . . T17), extracting those sub-areas (T1 . . . t17) and average
area values from the table, which are assigned to that combination
of radiation source position, detector element and image segment
with which the radiation source position and the detector element
determine the measured value (p.sub.j), on which the back
projection value (p.sub.j-{tilde over (p)}.sub.j) is dependent, for
which the proportionality factor (w.sub.ji) is to be determined,
and with which the image segment has the image value (c.sub.i), for
which the proportionality factor (w.sub.ji) is to be determined,
adding and/or subtracting the extracted sub-areas (T1 . . . t17),
so that the area of the detector element (51) results with which
the measured value has been taken up, on which the back projection
value (p.sub.j-{tilde over (p)}.sub.j) is dependent, and adding
and/or subtracting the associated, extracted average area values in
the same direction, while the result of the addition and/or
subtraction in the same direction is the proportionality factor
(w.sub.ji).
5. An imaging method as claimed in claim 1, characterized in that
several iterations are executed for the reconstruction of the
object image, until an abort criterion is satisfied, wherein,
initially the image values are set to an initial value and wherein
new image values (c.sup.n+1.sub.i) are determined in an iteration
with the following steps: forward projection along the rays
produced in step (a) through the object image, so that for each
measured value (p.sub.j) a fictitious intermediate measured value
({tilde over (p)}.sub.j) is produced from image values
(c.sup.n.sub.i), determining differences (p.sub.j-{tilde over
(p)}.sub.j) between the fictitious intermediate measured values
({tilde over (p)}.sub.j) and the measured values (p.sub.j) as back
projection values (p.sub.j-{tilde over (p)}.sub.j), back projection
of the differences (p.sub.j-{tilde over (p)}.sub.j) along the rays,
wherein a proportion of a difference (p.sub.j-{tilde over
(p)}.sub.j) is added to each image value ( )c.sup.n.sub.i and
wherein this proportion is dependent on the proportionality factor
(w.sub.ji).
6. An imaging method as claimed in claim 5, characterized in that
the proportion of a difference (p.sub.j-{tilde over (p)}.sub.j)
which is added to an image value (c.sup.n.sub.i), is proportional
to the respective proportionality factor (w.sub.ji).
7. A computer tomograph for executing the imaging method as claimed
in claim 1, comprising a reconstruction unit (10) for
reconstructing the spatial distribution of the absorption of the
object from the measured values acquired from the detector unit
(16) and a control unit (7) for controlling the radiation source
(S), the detector unit (16), the driving arrangement (2, 5) and the
reconstruction unit (10) in accordance with the steps of claim
1.
8. A computer program for a control unit for controlling a
radiation source (S), a detector unit (16), a driving arrangement
(2, 5) and a reconstruction unit (10) of a computer tomograph (1)
in accordance with the steps as claimed in claim 1.
Description
[0001] The invention relates to an imaging method, in particular a
computerized tomography method, in which an object is penetrated by
rays from different directions and with which measured values,
which depend on the intensity of the rays after penetrating the
object, are acquired with a detector unit. With the imaging method,
measured value-dependent back projection values are back projected
for the reconstruction of an object image, wherein the object image
is divided into spherically symmetric, overlapping image segments
(blobs). Moreover, the invention relates to a computer tomograph
for executing the method as well as a computer program for
operating the computer tomograph.
[0002] With an imaging method of the type mentioned above, back
projection values, which are, for example, equal to the measured
values, are back projected along a ray which, starting from the
radiation source, centrally strikes the respective detector
element. This type of back projection is disadvantageous, if the
rays emanating from the radiation source diverge, since the
spherically symmetric image segments, depending upon their distance
from the radiation source, are penetrated differently by the rays
during the back projection. An image segment that is relatively
close to the radiation source is penetrated by more rays than an
image segment that is at a greater distance from the radiation
source. This leads to problems, since for the reconstruction of a
good-quality image segment this image segment must be penetrated by
a determinable minimum number of rays during the back projection.
If this is not the case, then this is generally called "low
scanning" of the image segment and aliasing-artifacts become
visible in the object image.
[0003] Image segments, which are relatively remote from the
radiation source, are frequently penetrated by very few rays during
the back projection, so that aliasing-artifacts occur in the object
image. A possibility of suppressing these aliasing-artifacts is to
increase the size of the spherically symmetric image segments, so
that even the image segments relatively remote from the radiation
source are penetrated by a sufficient number of rays during the
back projection. However, this considerably reduces the resolution
of the object image.
[0004] It is therefore an object of the present invention to
provide an imaging method of the type mentioned in the opening
paragraph with which, even with divergent rays, object images with
high resolution and reduced noise are reconstructed by aliasing
artifacts and therewith, object images with an improved image
quality in comparison to the state of the art.
[0005] This object is achieved according to the invention by an
imaging method, particularly by a computerized tomography method,
comprising: [0006] Reconstruction of an object image by back
projection of back projection values that depend on measured
values, [0007] wherein the object image is divided into
overlapping, quasi-spherically symmetric image segments, each one
of which is defined by an image value and a quasi-spherically
symmetric base function, [0008] wherein the back projection values
are added in proportions to the image values during the back
projection.
[0009] The term "quasi-spherically symmetric image segments"
comprises both spherically symmetric image segments and image
segments that can be brought into a spherically symmetric form by a
linear coordinate transformation, for example, by a modification of
an axis scaling. Similarly holds for the term "quasi-spherically
symmetric base function".
[0010] With the imaging method according to the invention, back
projection values that are dependent on the measured values are
back projected, for example, a back projection value can be
proportional or equal to a measured value. During the back
projection a proportion of a back projection value is added to an
image value, while this proportion is dependent on the average
value of the line integrals of the base function belonging to the
respective image value along those rays that have generated the
measured value on which the respective back projection value is
dependent. In contrast, with the above-mentioned known method, this
proportion depends upon the line integral of the base function
belonging to the respective image value along the ray which,
starting from the respective radiation source, strikes the
respective detector element centrally, without an average value
being formed over several line integrals. This means that according
to the invention back projection is not effected along a ray, as
with the known method which, starting from the radiation source,
strikes the respective detector element centrally, but back
projection is effected along a beam, wherein this beam comprises
all rays that strike the respective detector element from the
radiation source. As a result, with the back projection, each
spherically symmetric image segment is penetrated by a larger
number of rays in comparison to the state of the art, so that the
aliasing artifacts are reduced.
[0011] With a back projection, an inverse measuring process is
simulated. With the measuring process itself, the intensity of the
beam is detected, whose rays strike the respective detector
element. With the known back projection method, the measured values
are then back projected along a ray that strikes the respective
detector element centrally starting from the radiation source. As a
result of this, the inverse measuring process is simulated only
insufficiently, since, as mentioned above, the detector element
does not detect individual rays but a beam during the measurement,
which beam comprises all the rays that strike the detector element.
With the back projection according to the invention, this beam is
taken into consideration in that the inverse measuring process is
simulated more realistically. This improved simulation of the
inverse measuring process, together with the reduction of aliasing
artifacts, leads to an improvement of the quality of the
reconstructed object image.
[0012] The imaging method in accordance with an embodiment, has a
proportional dependency between the proportion of a back projection
value, which is added to an image value during the back projection
and the proportionality factor. This leads to a further improvement
of the image quality.
[0013] In an embodiment, an imaging method is described, which
determines the proportionality factor with a low cost of
computation.
[0014] In the imaging method according to the invention and in
accordance with an embodiment, the object image is reconstructed
with an iterative method, which leads to a further improvement of
the image quality.
[0015] A computer tomograph for executing the method according to
the invention is described in claim 7. Claim 8 defines a computer
program for controlling the computer tomograph as claimed in claim
7.
[0016] The invention is further explained hereinafter with
reference to the drawings in which:
[0017] FIG. 1 shows a computer tomograph, with which the method
according to the invention can be executed,
[0018] FIG. 2 shows a flow chart of the method according to the
invention,
[0019] FIG. 3 shows a flow chart for generating sub-areas and
average area values, with which a proportionality factor can be
calculated at low computation cost,
[0020] FIG. 4 to FIG. 9 show different sub-areas on a detector area
and
[0021] FIG. 10 shows a flow chart of a back projection.
[0022] The computer tomograph represented in FIG. 1 comprises a
gantry 1, which can rotate around an axis of rotation 14 running
parallel to the z-direction of the coordinate system 22 represented
in FIG. 1. For this purpose, the gantry 1 is driven by a motor 2
with a preferably constant, but adjustable angular speed. A
radiation source S, for example, an X-ray device is fastened to the
gantry 1. This X-ray device is provided with a collimator
arrangement 3, which extracts a cone-shaped beam 4 from the
radiation generated by the radiation source S, that is, a beam that
has a finite expansion different from zero, both in z-direction and
in a direction perpendicular to it (that is, in a plane
perpendicular to the axis of rotation).
[0023] The beam 4 penetrates a cylinder-shaped examination area 13
in which an object, for example, a patient on an examination table
(neither is represented) or else a technical object can be located.
After passing through the examination area 13 the beam 4 strikes a
detector unit 16 with a detector area 18, which unit is fastened to
the gantry unit 1, which detector area comprises a multiplicity of
detector elements, which are arranged in rows and columns in a
matrix in the form of a rectangle in this embodiment. The detector
columns run parallel to the axis of rotation 14. The detector rows
are located in planes perpendicular to the axis of rotation, in
this embodiment on a circular arc around the radiation source S
(focus-centered detector area). In other embodiments they may,
however, be formed differently, for example, describe a circular
arc around the axis of rotation 14 or be straight. Each detector
element struck by the beam 4 provides a measured value for a ray
from the beam 4 in each position of the radiation source.
[0024] The beam angle of the beam 4 referred to as .alpha..sub.max
determines the diameter of the object cylinder, within which the
object to be examined is located when the measuring values are
acquired. The beam angle is then defined as the angle enclosed by a
ray located in a plane perpendicular to the axis of rotation 14 on
the edge of the beam 4, with a plane defined by the radiation
source S and the axis of rotation 14. The examination area 13, or
the object, or the examination table can be moved by means of a
motor 5 parallel to the axis of rotation 14 or to the z-axis.
Equivalently, however, also the gantry could be moved in this
direction. If it is a technical object and not a patient, the
object can be turned during an examination, while the radiation
source S and the detector unit 16 remain still.
[0025] If the motors 2 and 5 run simultaneously, the radiation
source S and the detector unit 16 describe a helix-shaped
trajectory 17 relative to the examination area 13. If, on the other
hand, the motor 5 for the feed in the direction of the axis of
rotation 14 stands still and the motor 2 causes the gantry to
rotate, there is a circular trajectory for the radiation source S
and the detector unit 16 relative to the examination area 13. In
the following example of embodiment only the helix-shaped
trajectory is considered.
[0026] The measured values acquired by the detector unit 16 are
applied to a reconstruction computer 10 (reconstruction unit),
which is connected to the detector unit 16, for example, by a
contactlessly operating data transmission (not represented). The
reconstruction computer 10 reconstructs the absorption distribution
in the examination area 13 and displays it, for example, on a
monitor 11. The two motors 2 and 5, the reconstruction computer 10,
the radiation source S and the transfer of the measured values from
the detector unit 16 to the reconstruction computer 10 are
controlled by a control unit 7.
[0027] The individual steps of an embodiment of the imaging method
according to the invention are explained hereinafter with reference
to the flow chart in FIG. 2.
[0028] After the initialization in step 101 the gantry rotates with
an angular speed, which is constant in this embodiment. But it can
also vary, for example, depending on the time or the position of
the radiation source.
[0029] In step 103 the examination area or the examination table is
moved parallel to the axis of rotation 14 and the radiation of the
radiation source S is switched on, so that the detector unit 16 can
detect the radiation from a multiplicity of angle positions and the
radiation source S moves relative to the examination area 13 along
the helix-shaped trajectory 17.
[0030] In step 105 an initial object image, divided into
spherically symmetric, overlapping image segments (blobs) is
predefined, each image segment having an image value and a
spherically symmetric base function. The image segments are
arranged in the object image at grid points of a three-dimensional,
cartesian grid. Such a division of the object image can be
represented by the following equation:
f ( x ) = i c i b i ( x ) ( 1 ) ##EQU00001##
[0031] Herein f(x) describes the object image, that is, if, for
example, an absorption distribution of the object is to be
reconstructed, f(x) represents the absorption value at the point x.
Furthermore, x.sub.i are the grid points at which the image
segments are arranged. Besides, c.sub.i is the image value and
b.sub.i(x) is the spherically symmetric base function of the image
segment arranged at the grid point x.sub.i. The image values are
initially equal to zero in this embodiment. The base functions
b.sub.i(x) are all equal here, so that b(x-x.sub.i) could be
substituted for b.sub.i(x) in equation (1).
[0032] As mentioned above, the image segments are arranged at grid
points of a three-dimensional, Cartesian grid. Alternatively, the
image segments may also be arranged at grid points of another type
of grid, for example, at grid points of a hexagonal, cubic
area-centered or cubic space-centered grid. Furthermore, the image
segments may also be arranged at grid points of a two-dimensional
grid, for example, if a two-dimensional object, like for example, a
layer of a three-dimensional object, is to be reconstructed.
[0033] The base function b.sub.i(x) is spherically symmetric.
Alternatively, it could also be arranged in a way that it can be
brought into a spherically symmetric form by a linear
transformation, for example, by an axial scaling.
[0034] In step 107 a table is provided, in which each combination
of radiation source position, image segment and detector element is
assigned one or more sub-areas of the detector area of the detector
unit 16 and average area values assigned to these sub-areas.
[0035] The sub-areas, which are assigned to a combination of
radiation source position, image segment and detector element, are
determined in accordance with the steps represented in FIG. 3.
[0036] First, in step 301, the base function of the image segment
is projected onto the detector area of the detector unit 16 along
the rays emanating from the radiation source position. That is, for
each ray emanating from the radiation source position, line
integrals of the base function of the image segment are formed
along the respective ray and the value of this line integral is put
at the point of impact of the respective ray on the detector
area.
[0037] Then, in step 303, the projection center of the base
function of the image segment is determined, that is, the position
of the projection of the center of the base function of the image
segment on the detector area.
[0038] In step 305 the sub-areas are determined. The sub-areas are
rectangular. They are selected in such a way that for each
sub-area, a corner coincides with the projection center of the base
function of the image segment determined in step 303 and that the
diagonally opposite corner of the respective sub-area coincides
with a corner of the detector element. This leads to the fact, that
sub-areas are determined for a combination of radiation source
position, image segment and detector element 1, 2 or 4, depending
on where the projection center of the base function of the image
segment is located relative to the detector element on the detector
area. This is further explained hereinafter in connection with the
FIGS. 4 to 9.
[0039] After the sub-areas have been determined for each
combination of radiation source position, image segment and
detector element, in step 307 for each sub-area, the average value
of the base function projected in step 301 is calculated of the
respective sub-area concerned, that is, the average value of the
line integrals determined in step 301 is calculated, which line
integrals are located on the sub-area concerned.
[0040] After one or more sub-areas and average area values have
been determined for each combination of radiation source position,
image segment and detector element and provided in a table, the
object image is iteratively reconstructed in the following steps.
One iteration step herein comprises the steps 109, 111 and 113.
[0041] In addition, in step 109 a forward projection is performed
for each combination of radiation source position and detector
element, which projection can be described by the following
equation:
p ~ j = i w ji c i n ( 2 ) ##EQU00002##
[0042] Herein, {tilde over (p)}.sub.j is a fictitious intermediate
measured value obtained by forward projection. The index j herein
features the respective combination of radiation source position
and detector element or the beam that is determined by this
combination. That is, for each combination of radiation source
position and detector element, {tilde over (p)}.sub.j defines the
intermediate measured value obtained by forward projection along
the rays emanating from the respective radiation source position
and striking the respective detector element. Moreover,
c.sup.n.sub.i defines the image value of the image segment arranged
at the point x.sub.i after the n-th iteration of the iterative
reconstruction method. Initially, the image values c.sub.0.sub.i
are set to zero. Alternatively, the image values c.sup.0.sub.i can
have other initial values. Furthermore, w.sub.ji describes a
proportionality factor, which indicates, to what proportions the
image values c.sup.n.sub.i are added for generating the
intermediate measured value {tilde over (p)}.sub.j.
[0043] For determining the proportionality factor w.sub.ji, the
base function b.sub.i(x) provided at the point x.sub.i for the
radiation source point of the j.sup.th combination of radiation
source position and detector element is projected onto the detector
area, as explained above in connection with step 301. Then the
average value of this projection is calculated from the area of the
detector element of the j.sup.th combination of radiation source
position and detector element, that is, the line integrals of the
base function b.sub.i(x) along the rays that strike the detector
element of the j.sup.th combination of detector element and
radiation source position, are averaged. The resulting average
value is the proportionality factor w.sub.ji.
[0044] The formation of the average values of line integrals is
effected preferably by arithmetic averaging.
[0045] With forward projection, a fictitious intermediate measured
value is determined for each measured value and thus for each
combination of radiation source position and detector element,
wherein the respective image values c.sup.n.sub.i are first
multiplied by the proportionality factor w.sub.ji and then added
together.
[0046] The proportionality factors w.sub.ji in this embodiment are
calculated with the help of the table provided in step 107.
[0047] The table contains one or more sub-areas and relevant
average area values for each combination of radiation source
position, image segment and detector element. For calculating the
proportionality factor w.sub.ji, the sub-areas and associated
average area values, which are assigned to the j.sup.th combination
of radiation source position and detector element and the image
segment at the point x.sub.i are inferred from the table. Then the
sub-areas inferred from the table are added together and/or
subtracted so that they are equal to the area of the detector
element of the j.sup.th combination of radiation source position
and detector element. The average area values assigned to the
sub-areas are then similarly added together and/or subtracted,
while the result of the similar addition and/or subtraction is the
respective proportionality factor. Similarly, addition and/or
subtraction is explained with reference to the FIGS. 4 to 9.
[0048] How the sub-areas and thus the associated average area
values are subtracted and/or added together, depends on the
position determined in step 303 of the projection center of the
base function b.sub.i(x) of the image segment at point x.sub.i
relative to the detector element of the j-th combination of
radiation source position and detector element.
[0049] This is represented hereinafter with reference to the
sections of the detector area 18 represented in the FIGS. 4 to 9.
In the FIGS. 7 to 9 the respective section of the detector area 18
is shown several times, in order to be able to represent
overlapping sub-areas separately from each other.
[0050] If the projection center 53 of the base function b.sub.i(x)
is located on the detector element 51 of the j.sup.th combination
of radiation source position and detector element and not on the
edge of the detector element 51 (FIG. 4), four sub-areas T1, T2,
T3, T4 can be inferred from the table, which are added together, in
order to obtain the area of the detector element 51. The
proportionality factor w.sub.ji would then be equal to the total of
the average area values assigned to the sub-areas T1, T2, T3,
T4.
[0051] If the projection center 53 of the base function b.sub.i(x)
is located on the edge of the detector element 51 of the j.sup.th
combination of radiation source position (FIG. 5), then two
sub-areas T5 and T6 can be inferred from the table, which are to be
added together, in order to obtain the area of the detector element
51. The proportionality factor w.sub.ji would then be equal to the
total of the average area values assigned to the sub-areas T5 and
T6.
[0052] If the projection center 53 of the base function b.sub.i(x)
of the i-th image value segment is located in a corner of the
detector element 51 of the j.sup.th combination of radiation source
position and detector element (FIG. 6), then a sub-area T11 can be
inferred from the table. In this case the average area value
assigned to this sub-area T11 is equal to the proportionality
factor w.sub.ji.
[0053] If the projection center 53 of the base function b.sub.i(x)
is not located on the detector element 51 of the j.sup.th
combination of radiation source position and detector element, but
in the same detector row or column and not on an edge of a detector
element (FIG. 7), then four sub-areas T7, T8, T9, T10 can be
inferred from the table. The area of the detector element 51
results from subtracting the two smaller sub-areas T9, T10 from the
total of the two larger sub-areas T7, T8 (T7+T8-T9-T10).
Correspondingly, the average area values assigned to these
sub-areas would then have to be added together or subtracted, in
order to obtain the proportionality factor w.sub.ji.
[0054] If the projection center 53 of the base function b.sub.i(x)
is not located on the detector element 51 of the j.sup.th
combination of radiation source position and detector element, but
in the same detector row or column and on an edge of a detector
element (FIG. 8), then two sub-areas T12, T13 can be inferred from
the table. The area of the detector element 51 results from
subtracting the sub-area T12 from the sub-area T13. For calculating
the proportionality factor w.sub.ji the average area value, which
is assigned to the sub-area T12, would in this case have to be
subtracted from the average area value that is assigned to the
sub-area T13.
[0055] If none of the aforementioned cases described in connection
with the FIGS. 4 to 8 is present, then four sub-areas T14, T15,
T16, T17 can be inferred from the table (FIG. 9). The largest
sub-area T17, for which the corner that is diagonally across from
the projection center 53 is farthest from the projection center 53,
and the smallest sub-area T14, for which the corner that is
diagonally across from the projection center 53 is closest to the
projection center 53, are added together and the remaining two
sub-areas T15, T16 are subtracted from the resulting total in order
to obtain the area of the detector element 51. For calculating the
proportionality factor w.sub.ji the average area values, which are
assigned to the sub-areas T14 and T17 would then have to be added
together and the average area values which are assigned to the
sub-areas T15 and T16 would have to be subtracted from the
resulting total.
[0056] In practice, the projection center 53 of the respective base
function will mostly not be located on the edge of a detector
element. Therefore, as a rule, four sub-areas are used for the
calculation of the proportionality factor.
[0057] After the forward projection, differences p.sub.j-{tilde
over (p)}.sub.j between the measured values p.sub.j and the
fictitious intermediate measured values {tilde over (p)}.sub.j are
formed as back projection values in step 111.
[0058] In step 113 the differences, that is, the back projection
values are back projected for each j.sup.th combination of
radiation source position and detector element and for each image
segment arranged in the point x.sub.i, in accordance with the
following equation:
c i n + 1 = c i n + .lamda. n p j - p ~ j i w ji 2 w ji ( 3 )
##EQU00003##
[0059] Herein, .lamda..sub.n is a proportionality factor of the
n.sup.th iteration. The proportionality factor .lamda..sub.n
determines, how strongly the image values c.sup.n.sub.i change from
one iteration step to the next and can optionally be predefined.
The proportionality factor .lamda..sub.n equals 1.
[0060] The back projection, in accordance with equation (3), can be
executed in accordance with the flow chart represented in FIG.
10.
[0061] Initially, in step 401 a point xi is determined in which no
image value c.sup.n.sub.i has yet been changed in this iteration
step.
[0062] Then, in step 403, a combination of radiation source
position and detector element is predefined, which combination has
not yet been used in this iteration step for the image value
c.sup.n.sub.i. This combination is again referred to by the index
j.
[0063] In step 405 the difference p.sub.j-{tilde over (p)}.sub.j
defined by the combination of radiation source position and
detector element selected in step 403, which difference represents
the back projection value, is multiplied by the proportionality
factor w.sub.ji and divided by a scale factor. In this example of
embodiment the scale factor is equal to the total of the squares of
those proportionality factors w.sub.ji that are assigned to the
combination of radiation source position and detector element
selected in step 403. If the proportionality factor .lamda..sub.n
in other examples of embodiment does not equal 1, then the
difference p.sub.j-{tilde over (p)}.sub.j is additionally
multiplied by .lamda..sub.n. The scaled difference multiplied by
the proportionality factor and if necessary by .lamda..sub.n, is
added to the image value c.sup.n.sub.i at the point x.sub.i.
[0064] In step 407 checks are made whether all combinations of
radiation source position and detector element, which led to a
measured value with the acquisition in step 103, have been
considered for the image value c.sup.n.sub.i in this iteration
step. If this is not the case, then step 403 is proceeded with.
[0065] Otherwise, in step 409 checks are made whether at all points
x.sub.i of the object image, the image values c.sup.n.sub.i have
been changed in this iteration step. If this is not the case, then
step 401 is proceeded with.
[0066] Otherwise the back projection of this iteration ends in step
411.
[0067] In step 115 checks are made whether an abort criterion is
satisfied. This abort criterion may be, for example, achieving a
predetermined number of runs of the iterative method. Moreover, the
abort criterion could be that the total of the square differences
of the fictitious intermediate values and of the measured
values
j ( p j - p ~ j ) 2 ##EQU00004##
falls below a predefined threshold value. If this abort criterion
is not satisfied, then step 109 is proceeded with. Otherwise the
imaging method ends in step 117.
[0068] The imaging method according to the invention is not limited
to iterative methods. The invention in fact comprises each imaging
method that uses a back projection for the reconstruction of an
object image, wherein when a back projection value depending on a
measured value is to be back projected, the proportionality of the
back projection value that is added to an image value is dependent
on the average value of the line integrals of the base function
belonging to the image value along those rays that have generated
the measured value on which the respective back projection value is
dependent. Therefore, the invention also comprises, for example,
imaging methods, which use a filtered back projection, which
methods take into account the proportion of a back projection value
that is added to an image value, as described above.
[0069] Moreover, the imaging method according to the invention is
not limited to the reconstruction of measured values, which were
produced with a computer tomograph. For generating the measured
values, each modality can be used that penetrates an object to be
represented by rays in different directions and acquires measured
values that depend on the intensity of the rays after penetrating
the object. Therefore, the measured values can also be produced
according to the invention with a C-Arc-System, a Positron Emission
Tomograph (PET) or a Single Photon Emission Computerized Tomography
(SPECT).
[0070] In the example of embodiment described, base functions are
projected onto the detector area and sub-functions and average
values are determined on this detector area. Alternatively, the
same views could be valid in any optional plane onto which the
detector elements and the base functions can be projected, whereas,
if the detector elements and the base functions are not found in
this plane, they must be projected into this plane.
LIST OF REFERENCE SIGNS
[0071] .alpha..sub.max Beam angle of the beam [0072] .lamda..sub.n
Proportionality factor [0073] b.sub.i(x) Spherical symmetrical base
function [0074] c.sub.i Image value of an image segment [0075]
c.sup.n.sub.i Image value of the i.sup.th image segment after the
n.sup.th iteration [0076] {tilde over (p)}.sub.j Fictitious
intermediate measured value [0077] p.sub.j Measured value [0078] T1
. . . T17 Sub-areas of the detector area [0079] w.sub.ji
Proportionality factor [0080] x.sub.i Grid points [0081] x Position
in object [0082] f(x) Object image value at point x [0083] S Source
of beams [0084] 1 Gantry [0085] 2, 5 Motor [0086] 3 Collimator
arrangement [0087] 4 Beam [0088] 7 Control unit [0089] 10
Reconstruction computer [0090] 11 Monitor [0091] 13 Area of
examination [0092] 14 Axis of rotation [0093] 16 Detector unit
[0094] 17 Helix-shaped trajectory [0095] 18 Detector area [0096] 22
Coordinate system [0097] 51 Area of a detector element [0098] 53
Projection center
* * * * *